reserve x,y,z,X for set,
  T for Universe;

theorem Th30:
  for T be non empty TopSpace, p be Point of T for x being Subset
  of T holds x in the carrier of OpenNeighborhoods p iff p in x & x is open
proof
  let T be non empty TopSpace, p be Point of T;
  let x be Subset of T;
  set Xp = { v where v is Subset of T: p in v & v is open };
  reconsider i = x as Subset of T;
  thus x in the carrier of OpenNeighborhoods p implies p in x & x is open
  proof
    assume x in the carrier of OpenNeighborhoods p;
    then ex v being Subset of T st i = v & p in v & v is open by Th29;
    hence thesis;
  end;
  assume p in x & x is open;
  then x in the carrier of InclPoset Xp;
  hence thesis by Th3;
end;
